Wind Energ. Sci., 6, 1061–1077, 2021https://doi.org/10.5194/wes-6-1061-2021© Author(s) 2021. This work is distributed underthe Creative Commons Attribution 4.0 License.

Vertical-axis wind-turbine computations usinga 2D hybrid wake actuator-cylinder model

Edgar Martinez-Ojeda1, Francisco Javier Solorio Ordaz1, and Mihir Sen2

1Facultad de Ingeniería, Universidad Nacional Autónoma de México, Mexico City, 04510, Mexico2Department of Aerospace and Mechanical Engineering,University of Notre Dame, Notre Dame, IN 46556, USA

Correspondence: Edgar Martinez-Ojeda (eamo87@protonmail.com)

Received: 7 May 2021 – Discussion started: 10 May 2021Revised: 8 July 2021 – Accepted: 19 July 2021 – Published: 16 August 2021

Abstract. The actuator-cylinder model was implemented in OpenFOAM by virtue of source terms in theNavier–Stokes equations. Since the stand-alone actuator cylinder is not able to properly model the wake of avertical-axis wind turbine, the steady incompressible flow solver simpleFoam provided by OpenFOAM wasused to resolve the entire flow and wakes of the turbines. The source terms are only applied inside a certainregion of the computational domain, namely a finite-thickness cylinder which represents the flight path of theblades. One of the major advantages of this approach is its implicitness – that is, the velocities inside the hollowcylinder region feed the stand-alone actuator-cylinder model (AC); this in turn computes the volumetric forcesand passes them to the OpenFOAM solver in order to be applied inside the hollow cylinder region. The processis repeated in each iteration of the solver until convergence is achieved. The model was compared against exper-imental works; wake deficits and power coefficients are used in order to assess the validity of the model. Overall,there is a good agreement of the pattern of the power coefficients according to the positions of the turbines inthe array. The actual accuracy of the power coefficient depends strongly on the solidity of the turbine (actuatorcylinder related) and both the inlet boundary turbulence intensity and turbulence length scale (RANS simulationrelated).

1 Introduction

The modeling of vertical-axis wind-turbine (VAWT) farmshas lacked researched in the last years compared tohorizontal-axis wind turbines (HAWTs). The complexity ofthe models ranges from simple momentum models to full-rotor RANS (Reynolds-averaged Navier–Stokes equations)or LES (large-eddy simulations) simulations. While simplemodels are computationally inexpensive, they lack accuracyand rely on various semi-empirical corrections which maynot be valid for all cases; high-fidelity simulations are sim-ply out of the scope of many researchers and scientists dueto the tremendous computational requirements.

This work proposes an actuator model integrated withinan OpenFOAM solver that relies on replacing the turbineby volumetric forces exerted on the fluid; this approacheliminates the need of highly resolved meshes around the

blades, thereby reducing the mesh size considerably. Theforces are modeled using the steady-state actuator-cylindermodel (Madsen, 1982), although any other model can be em-ployed. This approach is two-dimensional and the computa-tional time needed for simulating a wind farm ranges fromminutes to hours; moreover, the fidelity is superior to that ofsimple momentum models since the viscous wake is resolvedby the RANS simulation. This proposed model provides thecapability of serving as an optimization tool for vertical-axiswind-turbine farms. The models that have been used for windfarm modeling will be listed according to complexity; theirbenefits and caveats will be listed.

The first category belongs to models using the momentumtheory or potential flow theory. One of the simplest wakemodel is the Jensen model (Jensen, 1983), which is ableto describe the wake shape provided the thrust coefficient

Published by Copernicus Publications on behalf of the European Academy of Wind Energy e.V.

1062 E. Martinez-Ojeda et al.: Vertical-axis wind-turbine computations using a 2D hybrid wake actuator-cylinder model

and induction factors are given. The rate of growth of thewake depends on empirical constants. The model can be ex-tended to multiple turbines using a superposition technique.Although it was originally developed for HAWTs, some re-searchers have developed similar models for VAWTs (Abkar,2019), achieving good results for the wake shape. Potentialflow models try to emulate the wake of a turbine by superpos-ing many different potential flows such as a uniform stream,a dipole and a vortex (Whittlesey et al., 2010). The wakedeficit can be modeled as a probabilistic density function,and it is simply subtracted from the flow field. This model hasalso been extended to multiple-turbine environments (Arayaet al., 2014), but it is still very problem-dependent and needsmuch more calibration since it is unable to model viscouseffects.

The next category belongs to actuator models. As it wassaid before, these models rely on replacing the turbine byvolumetric forces. Svenning (2010) has successfully imple-mented a RANS actuator disc model for a HAWT in Open-FOAM. The model is explicit, and it requires the values ofthrust and torque; although it can be extended to multipleturbines, it relies on the assumption of all turbines having thesame thrust and torque, which may not be entirely true forturbines in the back rows. Bachant et al. (2018) have also de-veloped actuator line models for both HAWTs and VAWTswith the option of using either RANS or LES. Shamsod-din and Porté-Agel (2016), Abkar (2018) and Mendoza andGoude (2017) have also performed LES simulations of ac-tuator line models on a single VAWT. An interesting multi-turbine simulation using an actuator line model was done byHezaveh et al. (2018); the study works on the effect of clus-tering the turbines in order to increase the power density.

The last category of models employs full-rotor RANS sim-ulations. Works on multiple VAWTs can be found in Zan-forlin and Nishino (2016), Bremseth and Duraisamy (2016)and Giorgetti et al. (2015). Recently, a very interesting studyby Hansen et al. (2021) was done on pairs and triplets ofVAWTs; the study claims that a 15 % increase in power canbe achieved if turbines are placed closer. Although theseclaims are done on the basis of 2D simulations, it is not cer-tain whether this effect may scale to a wind farm.

The current RANS-AC has the potential of modeling en-tire wind farms without relying on empirical corrections forthe wake or without the need of HPC (high-performancecomputing). Moreover, only simple input data must be en-tered, namely the geometrical and operational parametersand inlet boundary conditions for the simulation.

2 Actuator-cylinder (AC) wake model for turbines

This section first presents the theory behind the AC model,then the justification of the linear solution is described in de-tail and a validation for the stand-alone AC is included. The

Figure 1. Radial forces acting on fluid.

last part deals with the details of the RANS-AC implementa-tion in OpenFOAM.

2.1 Stand-alone actuator-cylinder model

The VAWT can be modeled as a hollow cylinder upon whichradial volume forces fn(θ ) act. This will create a pressurejump across the entire surface (notice that the cylinder ismerely an abstraction; it does not exist materially). The tur-bine’s blades are responsible for these radial forces. Figure 1shows a cross section of an infinite long cylinder (in the z di-rection), the incoming wind velocity is V∞, 2ε is the cylin-der’s thickness and R is the radius. Madsen (1982) showedthat

Qn(θ )= limε→0

R+ε∫R−ε

fn(θ )dr, (1)

where Qn is the normal load per unit length exerted onthe fluid at angular position θ averaged over one revolu-tion 2πR. The AC coordinate system is x and y. The gov-erning equations are those of continuity and the steady-stateEuler equations. Velocities in the x and y directions are non-dimensionalized by the incoming wind velocity V∞; lengthsare non-dimensionalized by the wind-turbine radius R, andpressure is non-dimensionalized by ρV 2

∞.The Euler equations are applied to the entire field, and the

volume forces are represented by the forces exerted by theblades. The final form of the solution is shown in Eqs. (2a)and (2b), which allow calculation of the perturbation veloci-tieswx andwy ;N is the number of evaluation points, θ is theangle of the current evaluation point and φ is a dummy angleused for integration purposes. The turbine rotates counter-clockwise. This form only includes the linear part, althougha correction is made to make up for the non-linear terms.When the evaluation takes place inside the hollow cylinder,term I must be included; on the other hand, if the evaluation

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E. Martinez-Ojeda et al.: Vertical-axis wind-turbine computations using a 2D hybrid wake actuator-cylinder model 1063

point is located in the wake (the leeward part of the cylinder),both I and II must be included in the equation. Li (2017) pro-vides a useful discretization scheme assuming the forces arepiecewise constant.

wx =−1

2π

N−1∑i=0

Qn,i

θi+121θ∫

θi−121θ

−(x+ sinφ) sinφ+ (y− cosφ)cosφ(x+ sinφ)2+ (y− cosφ)2 dφ

−Qn(arccosy)︸ ︷︷ ︸I

+Qn(−arccosy)︸ ︷︷ ︸II

(2a)

wy =−1

2π

N−1∑i=0

Qn,i

θi+121θ∫

θi−121θ

−(x+ sinφ)cosφ− (y− cosφ) sinφ(x+ sinφ)2+ (y− cosφ)2 dφ (2b)

These equations can be put in matrix form using influence co-efficients that depend on geometrical variables that have to becomputed only once and a column vector Qn. Equation (2a)and (2b) can predict the perturbation velocities along the hol-low cylinder provided the forces are known, for which theblade element theory is needed. The free stream velocity V∞is broken down into its x and y Cartesian components so thatthey can be projected along the cylinder tangential and nor-mal directions. Note that α is the angle of attack with respectto the chord of the blade, ωR is the tangential velocity of theblade due to rotation, Vt and Vn are the airflow tangential andnormal velocities, and Vrel is the relative velocity. Sometimesthe chord is pitched slightly by an angle δ.

The normalization of the loads can be found in Li (2017),and it will be outlined briefly here for the case of acounterclockwise-rotating turbine. From now on, all vari-ables will be non-dimensional (written in lowercase). The lo-cal x and y velocities can be decomposed into the normal andtangential velocities.

vx = 1+wx, (3)vy = wy, (4)vn = vx sinθ − vy cosθ, (5)vt = vx cosθ + vy sinθ + λ (6)

The rotational speed of the rotor normalized by V∞ isωR/V∞, which is a common characteristic parameter ofwind turbines called the tip-speed ratio denoted by λ. Thenormalization of the relative speed and α are

vrel =

√v2

t + v2n, (7)

α = arctan(vn/vt)− δ. (8)

The aerodynamic forces are

Cn = CL cosα+CD sinα, (9)Ct = CL sinα−CD cosα. (10)

The lift and drag coefficients can be obtained from alookup table as a function of α and Reynolds number. Somecommon NACA profiles have plenty of experimental dataand can be found in Sheldahl and Klimas (1981) at highReynolds numbers and for an ample range of α. Finally thenormal and tangential forces exerted on the cylinder become

Qn(θ )=σ

2πv2

rel (Cn(θ )cosδ−Ct(θ ) sinδ) , (11)

Qt(θ )=−σ

2πv2

rel (Cn(θ ) sinδ+Ct(θ )cosδ) . (12)

The turbine σ is given by σ =NBc/2R, which can be in-terpreted as the blades’ area per unit length divided by theturbine swept area per unit length. It is important to keepσ low; otherwise the basic assumptions about the modelbreak down since effects such as flow curvature and flowdistortion are not taken into account. The model does notguarantee any results whatsoever if high-solidity rotors areused. Cheng (2016, p. 40) uses the AC model in his PhD the-sis, and the solidity values encountered there are fairly low– around 0.12 for a large two-bladed VAWT; Paraschivoiu(2002, p. 169) also employs low-solidity turbines in the de-velopment of his double-multiple stream-tube model, and thevalues of σ do not go beyond 0.22. The problem of solid-ity becomes important in small turbines as they have a largec/R ratio. According to Migliore et al. (1980), these bladesare subjected to a curvilinear flow which alters the bound-ary layer of the airfoil. Kinematic analysis from Miglioreet al. (1980) also shows that the angle of attack and the rela-tive wind velocity are dependent on the azimuthal angle, thetip-speed ratio and the chord-to-radius ratio; therefore α andvrel can vary significantly chordwise since any point on theblade has a unique radial distance. By employing conformalmapping techniques, it is possible to transform the airfoil inthe curvilinear flow to a virtual airfoil in a rectilinear flow.The transformation introduces a camber and an additionalangle of incidence – namely virtual camber and virtual inci-dence – which are also dependent on the azimuthal angle, al-though they can be averaged by the mean value of one revolu-tion. Thus it is shown in Migliore et al. (1980) that these vir-tual airfoils have lift at α = 0; therefore the CL vs. α curve isshifted upwards depending on the value of c/R. Not only thelift coefficient is affected but also the stall angle, which oc-curs much earlier as in the original airfoil without a camber;this premature stall deteriorates the efficiency of the wind tur-bine. Results from wind turbines with values of c/R = 0.114and c/R = 0.26 each in Migliore et al. (1980) show that thepower coefficient is strongly dwindled as c/R increases. Insummary, results from the AC model using relatively-high-solidity wind turbines will certainly miscalculate the angleof attack to a certain degree, thus overestimating the powercoefficient of the turbine.

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1064 E. Martinez-Ojeda et al.: Vertical-axis wind-turbine computations using a 2D hybrid wake actuator-cylinder model

The perturbation velocities can be determined if the forcesare known, while the forces also depend on the perturbationvelocities. The solution is iterative: first, the perturbation ve-locities are set to zero, then the aerodynamic coefficients arecomputed as well as Qn and Qt. Equations (2a) and (2b) areused to find the perturbation velocities, and the process is re-peated until convergence.

2.2 The linear correction

The aforementioned equations are only valid for low-loadedrotors (thrust coefficient), with the model including only thelinear part stops being accurate at high loads; however, arelation between the induction factor a and the thrust coef-ficient CT was found for the linear solution. Equating thisthrust coefficient to an empirical thrust coefficient from themomentum theory yields a correction factor for the inductionfactor at high loads. The perturbation velocities are all mul-tiplied by the correction factor ka. The procedure is straight-forward and can be found in Li (2017), Madsen et al. (2013),Ning (2016), Cheng et al. (2016) and Cheng (2016). Thereare, however, many empirical corrections. Madsen et al.(2013) provide the following equations for ka.

a = k3C3T+ k2C

2T+ k1CT+ k0, (13)

ka =

{1

1−a , a ≤ 0.151

1−a (0.65+ 0.35exp(−4.5(a− 0.15))), a > 0.15

},

(14)

where k3 = 0.0892, k2 = 0.0544, k1 = 0.251 and k0 =

−0.0017. Ning (2016) cites the following equations.

a =

12

(1−√

(1−CT)), CT ≤ 0.96

17

(1+ 3

√( 72CT− 3

)), CT > 0.96

(15)

ka =

{1

1−a , CT ≤ 0.9618a

7a2−2a+4 , CT > 0.96(16)

The thrust coefficient can be obtained using the followingequation, which can be found in Li (2017). This equation isvalid for counterclockwise and clockwise rotation.

CT =

2π∫0

(Qn(θ ) sinθ +Qt(θ )cosθ )dθ (17)

2.3 Validation against a 1.2 kW Windspire turbine

The Windspire was chosen for validation because it willbe used throughout this work; therefore the results of theAC model will be compared against experimental data. Ide-ally, it would be better to validate against a low-solidityturbine since it meets the requirements of the AC model;nevertheless, the Windspire turbine will be used despite ithaving a solidity of σ = 0.32. According to Zanforlin and

Figure 2. Lift polar for the DU06W200 airfoil at different Re.

Nishino (2016), the turbine is kept at an optimal tip-speed ra-tio λ= 2.3 up until 10.6 m s−1; after this point the rotationalspeed is kept constant and λ begins to decrease. The turbine’sradius is R = 0.6 m, the height is H = 6.1 m, the number ofblades is N = 3, the chord is c = 0.128 m and the airfoil isa DU06W200 section derived from a NACA0018 section,except the maximum thickness is 20 % and little camber isadded.

A particular challenge was to find polars for theDU06W200. Claessens (2006) provides both theoreticaland experimental data for Reynolds numbers of 300 000and 500 000 but does not give information whatsoever fora Reynolds number below 300 000; the turbine’s globalReynolds number at 8 m s−1 is Re = Rωc/ν = 130000, withω being the rotational speed. It was then decided to use po-lars from the software QBlade (Marten et al., 2013), whichis based on a vortex panel code derived from the MIT codeXfoil (Drela, 1989). QBlade is able to predict both drag andlift coefficients at angles of attack below stall; for rangesabove stall, an extrapolation can be done based on the Mont-gomerie extrapolation method, which is more accurate (atleast in this case) than the Viterna model. It was observedthat the Montgomerie model predicted better the shear dropin lift after stall has occurred. Figures 2 and 3 show the polarfor both the lift and drag coefficient at the typical Reynoldsnumbers encountered by this turbine; QBlade seems to havetrouble at low Reynolds numbers, and instabilities are mani-fested in the zone just before stall plus the fact that the slopebefore stall is not always linear and presents jagged seg-ments; despite the shortcomings, polars atRe = 50000Re =100000, Re = 150000 and Re = 200000 were included. Athigher Re it was found that QBlade overestimated lift andcould not predict well the shear drop of lift after stall accord-ing to wind tunnel data from Claessens (2006).

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E. Martinez-Ojeda et al.: Vertical-axis wind-turbine computations using a 2D hybrid wake actuator-cylinder model 1065

Figure 3. Drag polar for the DU06W200 airfoil at different Re.

No attempt was made to introduce dynamic stall or flowcurvature effects. Dynamic stall models can conflict with theAC model according to Li (2017). As for flow curvature dueto the turbine’s c/R = 0.21 – which is above 0.075 and 0.11in Paraschivoiu (2002, p. 169) and Cheng (2016, p. 40), re-spectively – it was decided not to employ any model due tothe increase in computational cost. Li (2017) uses Migliore’smodel, which computes the shape of a virtual airfoil with anadded camber (the original airfoil in a curved flow is mappedto a cambered airfoil in a straight flow); consequently, the liftand drag coefficients have to be recomputed according to theshape of the virtual airfoil – this needs models such as thevortex panel model, which can be expensive considering thatthe panel model has to be called for every azimuthal positiontimes the number of iterations.

The results from the AC model were compared againstdata from AC model results provided by Ning (2016); ex-perimental data are also available. Equations (15) and (16)were used for the linear correction. Figure 4 shows how bothAC models overpredict the CP. This overestimation must bein part because both AC models employ limited polars – e.g.,Ning (2016) using wind tunnel data at Re = 300000 and thiswork using data ranging from Re = 50000 to Re = 200000,thereby neglecting the fact that at lower wind speeds Re ismuch lower. The other reason must be because of the factthat the model is only two-dimensional, and no effects fromstruts, tower, tip losses, and flow curvature and dynamic stallare included. There is an overall good tendency; the resultsfrom the current work and the experimental data both peakat 10 m s−1. Without accurate polars from wind tunnel mea-surements, it is hard to get accurate results; the CP is there-fore very sensitive to the polars, and care must be taken wheninterpreting the lift and drag coefficients.

Figure 4. 1.2 kW Windspire turbine validation and comparison. λ iskept at 2.3 up until 10 m s−1.

2.4 RANS-AC implementation

The AC model can be incorporated into one of the Open-FOAM solvers by taking advantage of the source terms inthe Reynolds-averaged Navier–Stokes equations. The solverused is called simpleFoam (Moukalled et al., 2016), whichsolves the steady-state incompressible Navier–Stokes equa-tions for turbulent flows. A new solver called actuatorCylin-derSimpleFoam was made using the simpleFoam solver asa template. Whereas the solver needs minimal modification,the AC routines took most of the work. These routines areplaced in separate files. The k–ε turbulence model is pre-ferred since it has proven to yield relatively good results inenvironmental flows such as wakes (Bardina et al., 1997;Wilcox, 1998). Algorithm 1 shows the process followed bythe new solver.

Notice that N is the number of cylinders (turbines); eachcylinder has a set of corresponding cells where the velocityis read and then passed to the AC routine, which computesthe volumetric forces and passes them back to OpenFOAMusing the function Add Force. The thickness of the cylinder issubjective, and it will be explained in the next sections. Theway the volumetric forces are calculated by the RANS-ACis by assuming that they do not vary significantly across thethickness of the cylinder; therefore Eq. (1) becomes Eq. (18),where 1r is the thickness of the cylinder, and fn(θ ) are thevolumetric forces normal to the cylinder as a function of theazimuthal angle. These normal forces have to be projected inthe x and y direction of the volumetric field of the simulation.

fn(θ )=Qn(θ )/1r (18)

2.5 RANS-AC verification against AC model

In order to prove that the RANS-AC has been implementedcorrectly, the power coefficient of the RANS-AC will becompared against that of the stand-alone AC model. A sen-sitivity analysis concerning the thickness of the cylinder and

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1066 E. Martinez-Ojeda et al.: Vertical-axis wind-turbine computations using a 2D hybrid wake actuator-cylinder model

Table 1. Boundary conditions at time 0 for the computational domain.

Boundary conditions

U p volForce k ε νT

inlet freestream zeroGradient fixedValue freestream freestream calculatedoutlet zeroGradient fixedValue fixedValue freestream freestream calculatedtop/bottom freestream freestreamPressure fixedValue freestream freestream calculatedfront/back empty empty empty empty empty empty

the turbulence intensity I at the inlet of the domain will bediscussed. A uniform mesh was chosen for simplicity. Al-though OpenFOAM is provided with mesh refinement utili-ties, the refined mesh is inevitably three-dimensional due tothe meshing algorithm, which is even more computationalexpensive; therefore the refined mesh was discarded. Theboundary conditions are inlet, outlet, top and bottom, andback and front. Table 1 shows the boundary conditions forevery variable OpenFOAM has to compute.

It must be clear that the simpleFoam solver interprets pas p/ρ since the RANS equations are divided by ρ due to theflow being incompressible. Free stream conditions act like a

zero-gradient condition when the flow comes out of the do-main and act like a fixed value when it is not; it is a kindof inlet–outlet condition in case of having flow reversal. ThefreestreamPressure is an outlet–inlet condition that uses thevelocity orientation to act either as a zero-gradient conditionor a fixed-value condition. The empty boundary conditionmeans that nothing is calculated at those faces; this is onlyvalid for two-dimensional cases with one cell in the third di-rection.

In summary, all variables have to be initialized at time 0 inthe domain. In order to initialize the values of all variables, aset of equations is needed. Equations (19)–(22) are the turbu-

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Table 2. RANS-AC results from two different meshes verified against the stand-alone AC.

Mesh sensitivity results

Stand-alone AC Fine mesh Coarse mesh1.0 chord thickness cylinder 1.5 chord thickness cylinder50× 50 cell enclosing square 30× 30 cell enclosing square

U∞ CP CP CP I

4 0.22 0.196 0.197 0.1306 0.23 0.216 0.217 0.1258 0.26 0.275 0.270 0.13410 0.32 0.339 0.340 0.11612 0.25 0.230 0.230 0.11014 0.16 0.100 0.084 0.106

lence length scale, the turbulent kinetic energy, the turbulentkinetic energy dissipation rate and the turbulent viscosity, re-spectively. The least intuitive is l; this value is taken fromVersteeg and Malalasekera (1995, p. 66) based on the caseof a wake flow, where L is the wake width which will betaken as the diameter of the cylinder. Cm is just a constantset to 0.09 by default.

l = 0.08L (19)

k = 1.5(‖U‖I )2 (20)

ε =C0.75

m k1.5

l(21)

νT =Cmk

2

ε(22)

The RANS-AC was verified against the Windspire turbineusing a fine mesh and a coarse mesh. The mesh size wasbased on enclosing the cylinder in a n× n cell square; therest of the domain was meshed accordingly. Distances fromthe inlet to the turbine could range from 3 to 5D (diame-ters) as well as distances from the turbine to the sides. Dis-tances from the turbine to the outlet could be shorter than10D. No impact was observed in the CP. Table 2 shows thecomparison of the power coefficients as well as the meshparameters. No significant difference was observed betweenboth meshes, although both meshes underpredicted CP at14 m s−1. The number of time steps is dependent of the inletvelocity, e.g., the wake takes longer to develop when the in-let velocity is low. Although the wake development dependsstrongly on the inlet velocity and the value of ε, the power co-efficient reaches a stable value much earlier. This is verifiedin a log file. The development of the wake can be observedvisually by inspecting each time step; at 8 m s−1, 800 timesteps were sufficient to achieve the final shape of the wakeand a steady power coefficient.

The distance from the turbine to the outlet does not seemto affect the result. In this case, the outlet was placed 10Daway from the turbine. Care must be taken when choosingthe value of I ; data from Araya et al. (2014) were used to

Figure 5. Volumetric forces acting on the cylinder. Flow goes fromleft to right. Free stream speed is 8 m s−1, with a coarse mesh.

compute the value of I . Observations taken every 10 minfrom several wind directions and velocities were extracted;e.g., for 8 m s−1, data from velocities ranging from 7 to9 m s−1 were collected, and then the mean of the quotientof the standard deviation and the average velocity was calcu-lated. The same process was done for the rest of the velocitybins. A script added in Appendix A shows the procedure.

Since a volumetric field is created initially, at the end ofthe simulation it is possible to visualize these forces usingParaview. Figure 5 shows the volumetric forces acting on thecounterclockwise-rotating cylinder at 8 m s−1 (coarse mesh).It is reminded that the volumetric field is a vector field butthe magnitude is a scalar.

3 Validation against Araya et al. (2014)

This section is meant to test the capabilities of the RANS-AC in a multi-turbine environment. Experimental data froma small wind farm of VAWTs were found in Araya et al.(2014). These kinds of experiments are hard to find in the lit-erature since most experimental studies on VAWTs are doneon one turbine only. The experiment consists of a set of tur-bines that can be rearranged in any fashion in order to test

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1068 E. Martinez-Ojeda et al.: Vertical-axis wind-turbine computations using a 2D hybrid wake actuator-cylinder model

Figure 6. Four turbines in a row. The distance from turbine to tur-bine is 11.31D (diameters). Flow is from left to right.

Figure 7. Fish schooling configuration. Turbines are placed in acounterclockwise-rotating fashion. The flow is also from left toright.

the performance of several layouts; the location is in the An-telope Valley, California. The turbines are the same 1.2 kWWindspire turbines mentioned in the past sections. Althoughdata for multiple wind-turbine arrays are available, only datafrom two different arrays will be used here, namely an arrayof four turbines and another array of 18 turbines. Figures 6and 7 show the layouts of the two arrays. The 18-turbine ar-ray has counterclockwise-rotating turbines. It is said in Arayaet al. (2014) that the most prevalent wind direction is fromthe southwest; therefore the turbines in Figs. 6 and 7 werealigned in that direction. The wind speed is about 8 m s−1

and λ is kept at 2.3. The value of I is set to 0.13 according toTable 2.

3.1 Model calibration

Since the numerical simulation must be provided with theturbulence length scale in order to compute the turbulent ki-netic energy dissipation rate at the inlet, a study on the widthof the wake was conducted in order to find the appropriateturbulence length scale. The results in Table 1 were obtainedsupposing that the width of the wake was similar to the diam-eter of the turbines. This did not impact the results of the CP;

Figure 8. Wake width at different downwind stations. The widthbegins to reach a stable value at 7D, and its value is 2.8 m.

however, it was observed that the streamwise developmentof the wake was sensitive to this value – this is reflected in εsince it depends on l = 0.08L, where L is the width of thewake.

In order to obtain a value for l, a simulation with L=Dwas run and the width of the wake was found. Care must betaken in selecting the width of the wake as it varies down-stream. It was decided to take the value at 7D, roughly; thereason for this is that the rate of growth begins to reach asteady value. The rate of growth of the wake at small dis-tances downwind cannot be neglected. At larger distancesthe wake begins to fade away and a wake width is hard todefine. This is exemplified in Fig. 8; the procedure consistedin placing several downwind stations, e.g., crosswind plotsof the magnitude of the velocity. The width was then mea-sured from end to end, where each end has a velocity value ofthe free stream velocity, which is 8 m s−1 in this case. Theseends can be found visually by intersecting the wake plot witha horizontal line drawn at U = 8 m s−1, where U is the mag-nitude of the velocity. Notice in the figure that the locationat which the ends of the wake stop varying is at 7D approx-imately. The width is then y+− y−, where y+ is the upperend and y− is the lower end. It is also interesting to noticethe skewness of the wake, since the turbine is rotating coun-terclockwise; most of the power is extracted in the positiveportion of y.

Once the new value of L was found, an iterative proce-dure following the same logic was conducted: a new sim-ulation with L= 2.8 was conducted, and the width of thewake was obtained in the same fashion. The procedure was

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Figure 9. Wake widths obtained by following an iterative proce-dure. All the plots are located at 7D downwind.

stopped when the width stops varying across iterations. Fig-ure 9 shows the final value of the width of the wake, which is3.7 m; the turbulence length scale is found by substituting 3.7in l = 0.08L.

3.2 Array of four turbines

The power coefficients from this array were obtained fromdata published by Araya et al. (2014). The script used toextract information from the data set is presented in Ap-pendix B. The distance from turbine to turbine is 11.31D.Figure 10 shows the CP and the normalized CP. The latterwas taken to be the current turbine CP divided by the lead-ing turbine CP; experimental data were normalized with theleading turbine’s experimental CP, and numerical data werenormalized accordingly. There is a clear overestimation ofthe CP; as discussed earlier, the AC model tended to overes-timate the CP of this particular turbine. The normalized CPshows that there is an overall good trend: the power coeffi-cients decrease in the same manner.

Another plot concerning the velocity and turbulence in-tensity along the center line is included in Fig. 11. The mag-nitude of the velocity is normalized with respect to the freestream velocity U∞. The value of I was calculated in Par-aview by creating a new field according to the followingequation derived from Eq. (20).

I =

√(2/3)k‖U‖

(23)

The RANS-AC underestimates the wake recovery in betweenthe turbines. The value of I starts at 0.13 according to Ta-

ble 2, and then it reaches a steady pattern past the secondturbine; values up to 0.4 can be found near the wake of eachturbine.

3.3 Array of 18 turbines

A plot similar to Fig. 10 is presented for this case. Unfor-tunately, the CP’s across the array do not follow a coherentpattern; e.g., turbines that have been blocked present simi-lar or even higher CP’s than the turbines free of blockage.Figure 12 shows the current CP in panel (a) and the normal-ized CP in panel (b). The normalized CP was obtained bydividing each CP by the maximum CP.

It was observed that a portion of the angles of attack of theturbines that were free of blockage were above stall accord-ing to Fig. 2. This is wrong since the manufacturer states thatthe turbine is kept at an optimal λ of 2.3, therefore mean-ing that it is not stalled. As the flow traverses each turbinedownwind, it loses momentum; therefore each blocked tur-bine sees lower relative velocities and thus lower angles of at-tack. It would be intuitive to think that lower angles of attacklead to lower lift coefficients, but since the turbine is stalled,the lift coefficients might be even higher than those in thestalled regime. Data extracted from a row of turbines in thearray are presented in Fig. 13. The plot shows the angles ofattack and lift coefficients from turbines 2, 10 and 18. It canbe seen that there is indeed a decrease in the amplitude of αas each turbine presents blockage from another turbine; how-ever, it is important to notice that, in this case, turbine 18, forinstance, has the lowest amplitude of α, but its coefficientsare not in the stall region where they drop sharply, thereforeachieving higher lift and tangential force coefficients. The lo-cal Re ranges from 100 000 to 200 000, and the positive stallangle of attack is about 16◦ according to Fig. 2. Turbine 18has a maximum positive α of 13◦; therefore it operates at anoptimal regime, which should not happen actually.

It must be clear that this fault is due to the wrong pre-dictions of α in the AC model, which is possible in casehigh-solidity turbines are being used. The incoherent patternof CP’s along a row of turbines did not occur in the case ofthe array of four turbines. It is thought that this array wasnot affected by the accelerated flow in between turbines asin the case of the array of 18 turbines; therefore the velocityacross the center line decayed faster and the turbines in theback rows operated in a regime well below stall, achievingeven lower CP’s.

4 Verification with Shamsoddin andPorté-Agel (2016)

As seen in Sect. 3, the case of the 18-turbine wind farmdid not yield good results mainly because of the inabilityof the AC model to correctly predict angles of attack forhigh-solidity turbines. In seeing this, an additional verifica-tion study was done to prove that the RANS-AC does work

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1070 E. Martinez-Ojeda et al.: Vertical-axis wind-turbine computations using a 2D hybrid wake actuator-cylinder model

Figure 10. Power coefficients of the four-turbine array. Panel (a) is the actual CP, and panel (b) is the normalized CP.

Figure 11. Wake across the turbine array. The axis for I is located at the right side of the plot. Vertical dotted lines signify the locations ofeach turbine.

Figure 12. Power coefficients of the 18-turbine array.

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Figure 13. α and CL from turbines T02, T10 and T18.

Figure 14. Comparison of CP results from an LES simulation withthe current RANS-AC. The turbine’s σ is 0.09.

well indeed if low-solidity turbines are used. The work fromShamsoddin and Porté-Agel (2016) is used as a reference.An LES simulation was carried out on a three-bladed VAWTwith a radius of 25 m, a chord of 1.5 m and a height of100 m. The wing’s cross section is an NACA0018. The ro-tational speed is 16.5 rpm, and the wind speed is 9.6 m s−1,thus yielding a tip-speed ratio of 4.5. The turbulence inten-sity value at the inlet is 0.083, and an atmospheric boundarylayer is used, although this is not possible in a 2D simula-tion. Figure 14 shows a CP plot against λ. The current RANSsimulation was done a cylinder thickness of two chords andan enclosing square of 30× 30 cells. Although the RANS-AC model overpredicts the value of CP, there is a very goodagreement on the trend of the curve. Results from the stand-alone AC model are presented too.

Next, a comparison of the wake of the turbine is presented.The LES wake is taken at the Equator. The width of thewake L was taken as the diameter of the turbine, and theiterative procedure shown in Sect. 3 was not done since theresults using the diameter as the width of the wake were sat-isfactory. The accurate width of the wake seemed to impactmostly the initial value of ε (which depends on l = 0.08L) atthe inlet, and not much difference was observed in the wakerecovery when using the diameter as the width of the wake.Figure 15 shows very good agreement in the development ofthe wake.

Finally, the same 18-turbine wind farm simulation is done,except this time the turbine from Shamsoddin and Porté-Agel(2016) is used. The relative distances from turbine to turbineare preserved. The Reynolds number could not be preservedbecause that would imply running the turbine at extremelylow wind speeds and extremely low rpm. The purpose isto show that the RANS-AC has no trouble predicting theright trend of the power coefficients when using low-solidityturbines. The solver converged at 1564 iterations, althoughthe power coefficients reached an almost constant value at1000 iterations. Roughly an hour had passed by the time thesolver reached 1000 iterations; this was done on an all-in-one computer using only one 2.5 GHz processor and 12 GBRAM. The distance from the last turbine to the outlet washalf the distance from the first turbine to the last turbine, or12 (maxx −minx). Larger distances yielded the same results.It must be mentioned that it is not necessary to resolve theentire wake of the turbines until they recover the free streamvalue; this is possible thanks to the velocity outlet boundarycondition, which is zeroGradient. Figure 16 shows the powercoefficients and the normalized power coefficients. The nor-malized coefficients were calculated by dividing all CP’s by

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1072 E. Martinez-Ojeda et al.: Vertical-axis wind-turbine computations using a 2D hybrid wake actuator-cylinder model

Figure 15. Comparison of the LES wake at different downwind distances with the current RANS-AC results.

Figure 16. Wind farm array of 18 turbines using the 25 m radius turbine from Shamsoddin and Porté-Agel (2016). Panel (a) shows thecurrent CP, and panel (b) shows the normalized CP. Arrows pointing upwards denote turbines that are blocked by only one turbine, whereasarrows that point downwards denote turbines that have been blocked by two turbines.

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E. Martinez-Ojeda et al.: Vertical-axis wind-turbine computations using a 2D hybrid wake actuator-cylinder model 1073

the maximum CP. A coherent pattern is observed: the higherthe blockage, the lower the CP value is. Also, the CP’s of theturbines free of blockage were notably higher than those ofthe plot in Fig. 14; a maximum of 0.59 was found in one ofthe leading turbines, whereas a single isolated turbine had aCP of 0.49. It is believed that the accelerated flow in some re-gions impacts the CP’s, and thus different values are obtainedas if they were isolated.

5 Conclusions

The RANS-AC was successfully implemented in Open-FOAM. This was confirmed by the fact that it could achieve apower coefficient very similar to the stand-alone AC. Guide-lines for selecting the mesh size and the thickness of the ac-tuator were also given along with inlet boundary conditionsfor the RANS simulation. The model was validated againstmulti-turbine experiments, and good agreement was foundconcerning the trend of the power coefficients in a row offour VAWTs. Unfortunately these multi-turbine experimentswere done using small turbines which had a high solidity;this caused the model to wrongly predict the angles of at-tack, namely overestimating the angles of attack and thusgetting the wrong coefficients of lift and tangential force.Thus results from the array of 18 turbines could not matchthe experimental data. An additional verification against alarge VAWT LES simulation (Shamsoddin and Porté-Agel,2016) with a low solidity was conducted to prove that theRANS-AC is indeed capable of model the wind farm powercoefficients so long as the leading turbines were not incor-rectly predicted in the stall regime by the AC. The compar-ison with the results of the wake of the large VAWT was invery good agreement. Although a multiple-turbine simula-tion was not done in Shamsoddin and Porté-Agel (2016), asimulation of an array of 18 turbines preserving the originalrelative distances between turbines in Araya et al. (2014) wasconducted. This time, the RANS-AC predicted a coherentpattern of the power coefficients across the array; e.g., the tur-bines free of blockage had higher power coefficients, and theturbines that experienced more blockage from other turbineshad lower power coefficients. The RANS-AC can thereforebe used to model entire wind farms provided low-solidityturbines are used. Future work will consider wind farm opti-mizations with this model along with artificial intelligence topredict the performance of a particular wind farm in differentlocations. A neural network can be used to learn to predictthe power coefficients of a farm depending the wind veloc-ity and wind direction. This farm could then predict its ownperformance when seeing different wind speeds and direc-tions. Choosing the right array for a particular location canpotentially impact the rate of return of a wind farm project.

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1074 E. Martinez-Ojeda et al.: Vertical-axis wind-turbine computations using a 2D hybrid wake actuator-cylinder model

Appendix A: Turbulence intensity data script

import pandas as pddf = pd.read_csv(path, skiprows=5)df.head

data = df.iloc[:, [7, 8]]data.columns = ['avg', 'std']

# I at 4 m/sdf4 = data[(data.avg > 3) & (data.avg < 5)]

print( (df4['std'] / df4['avg']).mean() )

# I at 6 m/sdf6 = data[(data.avg > 5) & (data.avg < 7)]

print( (df6['std'] / df6['avg']).mean() )

# I at 8 m/sdf8 = data[(data.avg > 7) & (data.avg < 9)]

print( (df8['std'] / df6['avg']).mean() )

# I at 10 m/sdf10 = data[(data.avg > 9) & (data.avg < 11)]

print( (df10['std'] / df10['avg']).mean() )

# I at 12 m/sdf12 = data[(data.avg > 11) & (data.avg < 13)]

print( (df12['std'] / df12['avg']).mean() )

# I at 14 m/sdf14 = data[(data.avg > 13) & (data.avg < 15)]

print( (df14['std'] / df14['avg']).mean() )

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Appendix B: Power coefficients data script

import pandas as pddf = pd.read_csv(path, skiprows=5)df.head

data = df.iloc[:, [2, 4, 7, 11]]data.columns = ['n', 'p', 'u', 'd']

cp_02 = data[ (data['n'] == 2) &((data['u'] > 7.5) & (data['u'] < 8.5)) &

((data['d'] > 210) & (data['d'] < 235)) ]['p'].mean()

cp_10 = data[ (data['n'] == 10) &((data['u'] > 7.5) & (data['u'] < 8.5)) &

((data['d'] > 210) & (data['d'] < 235)) ]['p'].mean()

cp_18 = data[ (data['n'] == 18) &((data['u'] > 7.5) & (data['u'] < 8.5)) &

((data['d'] > 210) & (data['d'] < 235)) ]['p'].mean()

cp_24 = data[ (data['n'] == 24) &((data['u'] > 7.5) & (data['u'] < 8.5)) &

((data['d'] > 210) & (data['d'] < 235)) ]['p'].mean()

H = 6.1D = 1.2A = H * DU = 8RHO = 1.15pinf = (1 / 2) * RHO * (U ** 3) * A

cp_02 /= pinf # cp of turbine 02cp_10 /= pinf # cp of turbine 10cp_18 /= pinf # cp of turbine 18cp_24 /= pinf # cp of turbine 24

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1076 E. Martinez-Ojeda et al.: Vertical-axis wind-turbine computations using a 2D hybrid wake actuator-cylinder model

Code availability. Codes are available athttps://doi.org/10.5281/zenodo.5177216 (Martinez-Ojeda, 2021a).

Data availability. Data sets are available athttps://doi.org/10.5281/zenodo.5177214 (Martinez-Ojeda, 2021b).

Video supplement. This video explains the results of the 18-large-turbine wind farm. Pressure and velocity fields are shown(https://www.youtube.com/watch?v=Q6qZG6-oC3A, last access:10 August 2021) (Martinez-Ojeda, 2021c).

Supplement. The supplement related to this article is availableonline at: https://doi.org/10.5194/wes-6-1061-2021-supplement.

Author contributions. EMO was the main author of this work;he developed the stand-alone actuator-cylinder code and the actua-torCylinderSimpleFoam solver for OpenFOAM. MS contributed tothe guidance, suggestions and some of the writing in LaTeX as wellas some figures. FJSO contributed to the guidance, suggestions andrevision of this work.

Competing interests. The authors declare that they have no con-flict of interest.

Disclaimer. Publisher’s note: Copernicus Publications remainsneutral with regard to jurisdictional claims in published maps andinstitutional affiliations.

Acknowledgements. The main author of this work would like tothank CONACYT (Consejo Nacional de Ciencia y Tecnología) aswell as the OpenFOAM community.

Review statement. This paper was edited by Alessandro Bian-chini and reviewed by two anonymous referees.

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